Let ​f(x)equalsleft brace Start 2 By 2 Matrix 1st Row 1st Column 6 minus x comma 2nd Column x less than 3 2nd Row 1st Column x plus 2 comma 2nd Column x greater than 3. EndMatrix

Complete parts a through d below.
0
5
10
0
5
10
x
y
y equals 6 minus xy equals x plus 2

A coordinate system has a horizontal x-axis labeled from 0 to 10 in increments of 1 and a vertical y-axis labeled from 0 to 10 in increments of 1. A graph has two branches. The first branch is a ray that falls from left to right, starting at (6,0) and ending at an open dot at (3,3). The first branch is labeled y equals 6 minus x. The second branch is a ray that rises from left to right, starting at an open dot at (3, 5) and passing through (4, 6). The second branch is labeled y equals x plus 2.
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Part 1
a. Find ModifyingBelow lim With x right arrow 3 Superscript plus f left parenthesis x right parenthesis and ModifyingBelow lim With x right arrow 3 Superscript minus f left parenthesis x right parenthesis. Select the correct choice below​ and, if​ necessary, fill in any answer​ box(es) in your choice.

To solve the problem, we need to find the left-hand limit (\(\lim_{{x \to 3^-}} f(x)\)) and the right-hand limit (\(\lim_{{x \to 3^+}} f(x)\)) of the piecewise function \(f(x)\) as \(x\) approaches 3. 

The function \(f(x)\) is defined as:
\[
f(x) =
\begin{cases}
6 - x, & \text{if } x < 3 \\
x + 2, & \text{if } x > 3
\end{cases}
\]

### Step 1: Find the Left-Hand Limit (\(\lim_{{x \to 3^-}} f(x)\))
The left-hand limit is the value that \(f(x)\) approaches as \(x\) gets closer to 3 from the left side (\(x < 3\)). 

For \(x < 3\), the function is given by \(f(x) = 6 - x\). So,
\[
\lim_{{x \to 3^-}} f(x) = 6 - 3 = 3
\]

### Step 2: Find the Right-Hand Limit (\(\lim_{{x \to 3^+}} f(x)\))
The right-hand limit is the value that \(f(x)\) approaches as \(x\) gets closer to 3 from the right side (\(x > 3\)).

For \(x > 3\), the function is given by \(f(x) = x + 2\). So,
\[
\lim_{{x \to 3^+}} f(x) = 3 + 2 = 5
\]

### Conclusion
- The left-hand limit is \(\lim_{{x \to 3^-}} f(x) = 3\).
- The right-hand limit is \(\lim_{{x \to 3^+}} f(x) = 5\).

Since the left-hand limit and the right-hand limit are not equal, the overall limit \(\lim_{{x \to 3}} f(x)\) does not exist.

### Answer:
The correct choice is:
\[
\lim_{{x \to 3^-}} f(x) = 3 \quad \text{and} \quad \lim_{{x \to 3^+}} f(x) = 5
\]

Would you like to explore this further or see how the rest of the question is solved?

Here are five related questions that might interest you:

1. How do you find the overall limit of a piecewise function when the left-hand and right-hand limits are different?
2. What is the graphical interpretation of a limit at a point where the function is discontinuous?
3. How does the concept of one-sided limits relate to continuity at a point?
4. How would you find the limit at a point where the function is continuous?
5. What are some common situations where limits do not exist?

**Tip:** When analyzing piecewise functions, always consider the behavior of the function from both sides of the point in question to determine limits and continuity.

Learn how to find the left-hand and right-hand limits of the piecewise function f(x) as x approaches 3. Understand the graphical interpretation of the function with two branches: y = 6 - x for x < 3 and y = x + 2 for x > 3. Suitable for high school students studying piecewise functions and limits.

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